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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|84}}
{{ED intro}}


== Theory ==
== Theory ==
In the [[13-limit]] it is the [[optimal patent val]] for the rank five temperament tempering out [[144/143]].
84edo shares the [[3/2|perfect fifth]] with [[12edo]], [[tempering out]] the [[Pythagorean comma]] in its [[patent val]]. In the [[5-limit]] it tempers out the [[sensipent comma]]; in the [[7-limit]] [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[support]]ing [[orwell]], [[compton]], and [[sensei]]. In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]].  


84edo is where the [[orwell]] temperament takes its name from, since the generator of 7/6 is equal to 19 steps of the edo, referencing the [[Wikipedia:Nineteen Eighty-Four|book 1984]]. From a regular temperament perspective, orwell in 84edo comes in two varieties – the 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} representing [[newspeak]]. 84edo orwell offers mosses of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale.
84edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two varieties—the 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} supporting [[newspeak]]. 84edo orwell offers [[mos scale]]s of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale.
 
Being a small multiple of 12, 84et tempers out the [[Pythagorean comma]], thus supporting the period-12 temperament [[compton]]. Being a small multiple of 28, it tempers out the [[Oquatonic|oquatonic comma]], which maps 5/4 to 9\28.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|84|columns=14}}
{{Harmonics in equal|84|columns=12}}
{{Harmonics in equal|84|columns=10|start=15|collapsed=1|title=Approximation of prime harmonics in 84edo (continued)}}
{{Harmonics in equal|84|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 84edo (continued)}}


=== Miscellaneous properties ===
=== Subsets and supersets ===
84edo is a significantly composite number, with divisors of 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.  
84 is a [[largely composite]] number. Since 84 factors as {{factorization|84}}, 84edo has subset edos {{EDOs| 2, 3, 4, 6, 7, 12, 14, 21, 28, 42 }}. Being a small multiple of 28, it tempers out the [[oquatonic|oquatonic comma]], which maps 5/4 to 9\28.


== Table of intervals ==
== Intervals ==
For this table, the notation of Orwell[9] from the [[4L 5s]] page is taken. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & "amp" and a "at" (the symbol "@" unlike in the 4L 5s page cannot be used because of technical details).
{| class="wikitable center-1 right-2"
{| class="wikitable"
|-
|+Table of 84edo intervals
! #
! Degree
! Cents
! Size (Cents)
! Approximate ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and downs notation]]
! colspan="3" | [[4L 5s|4L 5s Notation]]
! Associated ratio
|-
|-
| 0
| 0
| 0.000
| 0.0
| [[1/1]]
| Perfect 1sn
| Perfect 1sn
| P1
| P1
| D
| D
| Perfect 1sn
| P1
| J
| 1/1 exact
|-
|-
| 1
| 1
| 14.286
| 14.3
| ''[[81/80]]'', [[105/104]], [[126/125]], [[169/168]], [[196/195]]
| Up 1sn
| Up 1sn
| ^1
| ^1
| ^D
| ^D
| Up 1sn
| ^1
| J^
|
|-
|-
| 2
| 2
| 28.571
| 28.6
| [[50/49]], [[64/63]], [[65/64]], ''[[91/90]]''
| Dup 1sn
| Dup 1sn
| ^^1
| ^^1
| ^^D
| ^^D
| Downaug 1sn
| vA1
| Jv&
|
|-
|-
| 3
| 3
| 42.857
| 42.9
| [[36/35]], [[40/39]], [[46/45]], [[49/48]]
| Trup 1sn
| Trup 1sn
| ^^^1
| ^^^1
| ^^^D
| ^^^D
| Aug 1sn
| A1
| J&
|
|-
|-
| 4
| 4
| 57.143
| 57.1
| ''[[27/26]]''
| Trudminor 2nd
| Trudminor 2nd
| vvvm2
| vvvm2
| vvvEb
| vvvEb
| Upaug 1sn, Downdim 2nd
| ^A1, vd2
| J^&, Kvaa
|
|-
|-
| 5
| 5
| 71.429
| 71.4
| [[24/23]], [[25/24]], [[26/25]], ''[[28/27]]''
| Dudminor 2nd
| Dudminor 2nd
| vvm2
| vvm2
| vvEb
| vvEb
| Dim 2nd
| d2
| Kaa
|
|-
|-
| 6
| 6
| 85.714
| 85.7
| [[20/19]], [[21/20]]
| Downminor 2nd
| Downminor 2nd
| vm2
| vm2
| vEb
| vEb
| Updim 2nd
| ^d2
| K^aa
|
|-
|-
| 7
| 7
| 100.000
| 100.0
| [[19/18]]
| Minor 2nd
| Minor 2nd
| m2
| m2
| Eb
| Eb
| Downminor 2nd
| vm2
| Kva
|
|-
|-
| 8
| 8
| 114.286
| 114.3
| [[15/14]], [[16/15]]
| Upminor 2nd
| Upminor 2nd
| ^m2
| ^m2
| ^Eb
| ^Eb
| Minor 2nd
| m2
| Ka
|
|-
|-
| 9
| 9
| 128.571
| 128.6
| [[14/13]]
| Dupminor 2nd
| Dupminor 2nd
| ^^m2
| ^^m2
| ^^Eb
| ^^Eb
| Upminor 2nd
| ^m2
| K^a
|
|-
|-
| 10
| 10
| 142.857
| 142.9
| [[13/12]]
| Trupminor 2nd
| Trupminor 2nd
| ^^^m2
| ^^^m2
| ^^^Eb
| ^^^Eb
| Downmajor 2nd
| vM2
| Kv
|
|-
|-
| 11
| 11
| 157.143
| 157.1
| [[23/21]]
| Trudmajor 2nd
| Trudmajor 2nd
| vvvM2
| vvvM2
| vvvE
| vvvE
| Major 2nd
| M2
| K
|
|-
|-
| 12
| 12
| 171.429
| 171.4
| [[21/19]]
| Dudmajor 2nd
| Dudmajor 2nd
| vvM2
| vvM2
| vvE
| vvE
| Upmajor 2nd
| ^M2
| K^
|
|-
|-
| 13
| 13
| 185.714
| 185.7
| [[10/9]]
| Downmajor 2nd
| Downmajor 2nd
| vM2
| vM2
| vE
| vE
| Downaug 2nd
| vA2
| Kv&
|
|-
|-
| 14
| 14
| 200.000
| 200.0
| [[9/8]]
| Major 2nd
| Major 2nd
| M2
| M2
| E
| E
| Aug 2nd
| A2
| K&
|
|-
|-
| 15
| 15
| 214.286
| 214.3
| [[26/23]]
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
| ^E
| ^E
| Upaug 2nd, Downdim 3rd
| ^A2, vd3
| K^&, Lva
|
|-
|-
| 16
| 16
| 228.571
| 228.6
| [[8/7]]
| Dupmajor 2nd
| Dupmajor 2nd
| ^^M2
| ^^M2
| ^^E
| ^^E
| Dim 3rd
| d3
| La
|
|-
|-
| 17
| 17
| 242.857
| 242.9
| [[15/13]], [[23/20]]
| Trupmajor 2nd
| Trupmajor 2nd
| ^^^M2
| ^^^M2
| ^^^E
| ^^^E
| Updim 3rd
| ^d3
| L^a
|
|-
|-
| 18
| 18
| 257.143
| 257.1
| [[52/45]]
| Trudminor 3rd
| Trudminor 3rd
| vvvm3
| vvvm3
| vvvF
| vvvF
| Down 3rd
| v3
| Lv
|
|-
|-
| 19
| 19
| 271.429
| 271.4
| [[7/6]]
| Dudminor 3rd
| Dudminor 3rd
| vvm2
| vvm2
| vvF
| vvF
| Perfect 3rd
| P3
| L
| [[7/6]]
|-
|-
| 20
| 20
| 285.714
| 285.7
| [[45/38]], [[46/39]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
| vF
| vF
| Up 3rd
| ^3
| L^
|
|-
|-
| 21
| 21
| 300.000
| 300.0
| [[19/16]], [[25/21]], [[32/27]]
| Minor 3rd
| Minor 3rd
| m3
| m3
| F
| F
| Downaug 3rd
| vA3
| Lv&
|
|-
|-
| 22
| 22
| 314.286
| 314.3
| [[6/5]]
| Upminor 3rd
| Upminor 3rd
| ^m3
| ^m3
| ^F
| ^F
| Aug 3rd
| A3
| L&
|
|-
|-
| 23
| 23
| 328.571
| 328.6
| [[23/19]]
| Dupminor 3rd
| Dupminor 3rd
| ^^m3
| ^^m3
| ^^F
| ^^F
| Upaug 3rd, Downdim 4th
| ^A3, vd4
| L^&, Mvaa
|
|-
|-
| 24
| 24
| 342.857
| 342.9
| [[28/23]], [[39/32]]
| Trupminor 3rd
| Trupminor 3rd
| ^^^m3
| ^^^m3
| ^^^F
| ^^^F
| Dim 4th
| d4
| Maa
|
|-
|-
| 25
| 25
| 357.143
| 357.1
| [[16/13]]
| Trudmajor 3rd
| Trudmajor 3rd
| vvvM3
| vvvM3
| vvvF#
| vvvF#
| Updim 4th
| ^d4
| M^aa
|
|-
|-
| 26
| 26
| 371.429
| 371.4
| [[26/21]]
| Dudmajor 3rd
| Dudmajor 3rd
| vvM3
| vvM3
| vvF#
| vvF#
| Downminor 4th
| vm4
| Mva
|
|-
|-
| 27
| 27
| 385.714
| 385.7
| [[5/4]]
| Downmajor 3rd
| Downmajor 3rd
| vM3
| vM3
| vF#
| vF#
| Minor 4th
| m4
| Ma
|
|-
|-
| 28
| 28
| 400.000
| 400.0
| [[24/19]]
| Major 3rd
| Major 3rd
| M3
| M3
| F#
| F#
| Upminor 4th
| ^m4
| M^a
|
|-
|-
| 29
| 29
| 414.286
| 414.3
| [[19/15]]
| Upmajor 3rd
| Upmajor 3rd
| ^M3
| ^M3
| ^F#
| ^F#
| Downmajor 4th
| vM4
| Mv
|
|-
|-
| 30
| 30
| 428.571
| 428.6
| [[9/7]], [[23/18]], [[32/25]]
| Dupmajor 3rd
| Dupmajor 3rd
| ^^M3
| ^^M3
| ^^F#
| ^^F#
| Major 4th
| M4
| M
|
|-
|-
| 31
| 31
| 442.857
| 442.9
| [[84/65]]
| Trupmajor 3rd
| Trupmajor 3rd
| ^^^M3
| ^^^M3
| ^^^F#
| ^^^F#
| Upmajor 4th
| ^M4
| M^
|
|-
|-
| 32
| 32
| 457.143
| 457.1
| [[13/10]], [[30/23]]
| Trud 4th
| Trud 4th
| vvv4
| vvv4
| vvvG
| vvvG
| Downaug 4th
| vA4
| Mv&
|
|-
|-
| 33
| 33
| 471.429
| 471.4
| [[21/16]]
| Dud 4th
| Dud 4th
| vv4
| vv4
| vvG
| vvG
| Aug 4th
| A4
| M&
|
|-
|-
| 34
| 34
| 485.714
| 485.7
| [[65/49]]
| Down 4th
| Down 4th
| v4
| v4
| vG
| vG
| Downminor 5th
| vm5
| Nva
|
|-
|-
| 35
| 35
| 500.000
| 500.0
| [[4/3]]
| Perfect 4th
| Perfect 4th
| P4
| P4
| G
| G
| Minor 5th
| m5
| Na
|
|-
|-
| 36
| 36
| 514.286
| 514.3
| [[27/20]]
| Up 4th
| Up 4th
| ^4
| ^4
| ^G
| ^G
| Upminor 5th
| ^m5
| N^a
|
|-
|-
| 37
| 37
| 528.571
| 528.6
| [[19/14]]
| Dup 4th
| Dup 4th
| ^^4
| ^^4
| ^^G
| ^^G
| Downmajor 5th
| vM5
| Nv
|
|-
|-
| 38
| 38
| 542.857
| 542.9
| [[26/19]]
| Trup 4th
| Trup 4th
| ^^^4
| ^^^4
| ^^^G
| ^^^G
| Major 5th
| M5
| N
| [[11/8]] in the 84b val
|-
|-
| 39
| 39
| 557.143
| 557.1
| [[18/13]]
| Trudaug 4th
| Trudaug 4th
| vvvA4
| vvvA4
| vvvG#
| vvvG#
| Upmajor 5th
| ^M5
| N^
|
|-
|-
| 40
| 40
| 571.429
| 571.4
| [[25/18]], [[32/23]]
| Dudaug 4th
| Dudaug 4th
| vvA4
| vvA4
| vvG#
| vvG#
| Downaug 5th
| vA5
| Nv&
|
|-
|-
| 41
| 41
| 585.714
| 585.7
| [[7/5]]
| Downaug 4th
| Downaug 4th
| vA4
| vA4
| vG#
| vG#
| Aug 5th
| A5
| N&
|
|-
|-
| 42
| 42
| 600.000
| 600.0
| [[27/19]], [[38/27]]
| Aug 4th, Dim 5th
| Aug 4th, Dim 5th
| A4, d5
| A4, d5
| G#, Ab
| G#, Ab
| Upaug 5th, Downdim 6th
| ^A5, vd6
| N^&, Ovaa
|
|-
|-
| 43
| 43
| 614.286
| 614.3
| [[10/7]]
| Updim 5th
| Updim 5th
| ^d5
| ^d5
| ^Ab
| ^Ab
| Dim 6th
| d6
| Oaa
|
|-
|-
| 44
| 44
| 628.571
| 628.6
| [[23/16]], [[36/25]]
| Dupdim 5th
| Dupdim 5th
| ^^d5
| ^^d5
| ^^Ab
| ^^Ab
| Updim 6th
| ^d6
| O^aa
|
|-
|-
| 45
| 45
| 642.857
| 642.9
| [[13/9]]
| Trupdim 5th
| Trupdim 5th
| ^^^d5
| ^^^d5
| ^^^Ab
| ^^^Ab
| Downminor 6th
| vm6
| Ova
|
|-
|-
| 46
| 46
| 657.143
| 657.1
| [[19/13]]
| Trud 5th
| Trud 5th
| vvv5
| vvv5
| vvvA
| vvvA
| Minor 6th
| m6
| Oa
|
|-
|-
| 47
| 47
| 671.429
| 671.4
| [[28/19]]
| Dud 5th
| Dud 5th
| vv5
| vv5
| vvA
| vvA
| Upminor 6th
| ^m6
| O^a
|
|-
|-
| 48
| 48
| 685.714
| 685.7
| [[40/27]]
| Down 5th
| Down 5th
| v5
| v5
| vA
| vA
| Downmajor 6th
| vM6
| Ov
|
|-
|-
| 49
| 49
| 700.000
| 700.0
| [[3/2]]
| Perfect 5th
| Perfect 5th
| P5
| P5
| A
| A
| Major 6th
| M6
| O
| [[3/2]]
|-
|-
| 50
| 50
| 714.286
| 714.3
| [[98/65]]
| Up 5th
| Up 5th
| ^5
| ^5
| ^A
| ^A
| Upmajor 6th
| ^M6
| O^
|
|-
|-
| 51
| 51
| 728.571
| 728.6
| [[32/21]]
| Dup 5th
| Dup 5th
| ^^5
| ^^5
| ^^A
| ^^A
| Dim 7th
| d7
| Paa
|
|-
|-
| 52
| 52
| 742.857
| 742.9
| [[20/13]], [[23/15]]
| Trup 5th
| Trup 5th
| ^^^5
| ^^^5
| ^^^A
| ^^^A
| Aug 6th
| A6
| O&
|
|-
|-
| 53
| 53
| 757.143
| 757.1
| [[65/42]]
| Trudminor 6th
| Trudminor 6th
| vvvm6
| vvvm6
| vvvBb
| vvvBb
| Downminor 7th
| vm7
| Pva
|
|-
|-
| 54
| 54
| 771.429
| 771.4
| [[14/9]], [[25/16]], [[36/23]]
| Dudminor 6th
| Dudminor 6th
| vvm6
| vvm6
| vvBb
| vvBb
| Minor 7th
| m7
| Pa
|
|-
|-
| 55
| 55
| 785.714
| 785.7
| [[30/19]]
| Downminor 6th
| Downminor 6th
| vm6
| vm6
| vBb
| vBb
| Upminor 7th
| ^m7
| P^a
|
|-
|-
| 56
| 56
| 800.000
| 800.0
| [[19/12]]
| Minor 6th
| Minor 6th
| m6
| m6
| Bb
| Bb
| Downmajor 7th
| vM7
| Pv
|
|-
|-
| 57
| 57
| 814.286
| 814.3
| [[8/5]]
| Upminor 6th
| Upminor 6th
| ^m6
| ^m6
| ^Bb
| ^Bb
| Major 7th
| M7
| P
| [[5/3]]
|-
|-
| 58
| 58
| 828.571
| 828.6
| [[21/13]]
| Dupminor 6th
| Dupminor 6th
| ^^m6
| ^^m6
| ^^Bb
| ^^Bb
| Upmajor 7th
| ^M7
| P^
|
|-
|-
| 59
| 59
| 842.857
| 842.9
| [[13/8]]
| Trupminor 6th
| Trupminor 6th
| ^^^m6
| ^^^m6
| ^^^Bb
| ^^^Bb
| Downaug 7th
| vA7
| Pv&
|
|-
|-
| 60
| 60
| 857.143
| 857.1
| [[23/14]], [[64/39]]
| Trudmajor 6th
| Trudmajor 6th
| vvvM6
| vvvM6
| vvvB
| vvvB
| Aug 7th
| A7
| P&
| [[105/64]]
|-
|-
| 61
| 61
| 871.429
| 871.4
| [[38/23]]
| Dudmajor 6th
| Dudmajor 6th
| vvM6
| vvM6
| vvB
| vvB
| Upaug 7th, Downdim 8th
| ^A7, vd8
| P^&, Qvaa
|
|-
|-
| 62
| 62
| 885.714
| 885.7
| [[5/3]]
| Downmajor 6th
| Downmajor 6th
| vM6
| vM6
| vB
| vB
| Dim 8th
| d8
| Qaa
|
|-
|-
| 63
| 63
| 900.000
| 900.0
| [[32/19]], [[27/16]], [[42/25]]
| Major 6th
| Major 6th
| M6
| M6
| B
| B
| Updim 8th
| ^d8
| Q^aa
|
|-
|-
| 64
| 64
| 914.286
| 914.3
| [[39/23]], [[76/45]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
| ^B
| ^B
| Down 8th
| v8
| Qva
|
|-
|-
| 65
| 65
| 928.571
| 928.6
| [[12/7]]
| Dupmajor 6th
| Dupmajor 6th
| ^^M6
| ^^M6
| ^^B
| ^^B
| Perfect 8th
| P8
| Qa
|
|-
|-
| 66
| 66
| 942.857
| 942.9
| [[45/26]]
| Trupmajor 6th
| Trupmajor 6th
| ^^^M6
| ^^^M6
| ^^^B
| ^^^B
| Up 8th
| ^8
| Q^a
|
|-
|-
| 67
| 67
| 957.143
| 957.1
| [[26/15]], [[40/23]]
| Trudminor 7th
| Trudminor 7th
| vvvm7
| vvvm7
| vvvC
| vvvC
| Downaug 8th
| vA8
| Qv
|
|-
|-
| 68
| 68
| 971.429
| 971.4
| [[7/4]]
| Dudminor 7th
| Dudminor 7th
| vvm7
| vvm7
| vvC
| vvC
| Aug 8th
| A8
| Q
|
|-
|-
| 69
| 69
| 985.714
| 985.7
| [[23/13]]
| Downminor 7th
| Downminor 7th
| vm7
| vm7
| vC
| vC
| Upaug 8th, Downdim 9th
| ^A8, vd9
| Q^, Rvaa
|
|-
|-
| 70
| 70
| 1000.000
| 1000.0
| [[16/9]]
| Minor 7th
| Minor 7th
| m7
| m7
| C
| C
| Dim 9th
| d9
| Raa
|
|-
|-
| 71
| 71
| 1014.286
| 1014.3
| [[9/5]]
| Upminor 7th
| Upminor 7th
| ^m7
| ^m7
| ^C
| ^C
| Updim 9th
| ^d9
| R^aa
|
|-
|-
| 72
| 72
| 1028.571
| 1028.6
| [[38/21]]
| Dupminor 7th
| Dupminor 7th
| ^^m7
| ^^m7
| ^^C
| ^^C
| Downminor 9th
| vm9
| Rva
|
|-
|-
| 73
| 73
| 1042.857
| 1042.9
| [[42/23]]
| Trupminor 7th
| Trupminor 7th
| ^^^m7
| ^^^m7
| ^^^C
| ^^^C
| Minor 9th
| m9
| Ra
|
|-
|-
| 74
| 74
| 1057.143
| 1057.1
| [[24/13]]
| Trudmajor 7th
| Trudmajor 7th
| vvvM7
| vvvM7
| vvvC#
| vvvC#
| Upminor 9th
| ^m9
| R^a
|
|-
|-
| 75
| 75
| 1071.429
| 1071.4
| [[13/7]]
| Dudmajor 7th
| Dudmajor 7th
| vvM7
| vvM7
| vvC#
| vvC#
| Downmajor 9th
| vM9
| Rv
|
|-
|-
| 76
| 76
| 1085.714
| 1085.7
| [[15/8]], [[28/15]]
| Downmajor 7th
| Downmajor 7th
| vM7
| vM7
| vC#
| vC#
| Major 9th
| M9
| R
|
|-
|-
| 77
| 77
| 1100.000
| 1100.0
| [[36/19]]
| Major 7th
| Major 7th
| M7
| M7
| C#
| C#
| Upmajor 9th
| ^M9
| R^
|
|-
|-
|-
| 78
| 78
| 1114.286
| 1114.3
| [[19/10]], [[40/21]]
| Upmajor 7th
| Upmajor 7th
| ^M7
| ^M7
| ^C#
| ^C#
| Downaug 9th
| vA9
| Rv&
|
|-
|-
| 79
| 79
| 1128.571
| 1128.6
| [[23/12]], [[25/13]], ''[[27/14]]'', [[48/25]]
| Dupmajor 7th
| Dupmajor 7th
| ^^M7
| ^^M7
| ^^C#
| ^^C#
| Aug 9th
| A9
| R&
|
|-
|-
| 80
| 80
| 1142.857
| 1142.9
| ''[[52/27]]''
| Trupmajor 7th
| Trupmajor 7th
| ^^^M7
| ^^^M7
| ^^^C#
| ^^^C#
| Upaug 9th, Downdim 10th
| ^A9, vd10
| R^&, Jva
|
|-
|-
| 81
| 81
| 1157.143
| 1157.1
| [[35/18]], [[39/20]], [[96/49]]
| Trud 8ve
| Trud 8ve
| vvv8
| vvv8
| vvvD
| vvvD
| Dim 10th
| d10
| Ja
|
|-
|-
| 82
| 82
| 1171.429
| 1171.4
| [[45/23]], [[49/25]], [[63/32]], [[128/65]], ''[[180/91]]''
| Dud 8ve
| Dud 8ve
| vv8
| vv8
| vvD
| vvD
| Updim 10th
| ^d10
| J^a
|
|-
|-
| 83
| 83
| 1185.714
| 1185.7
| Down 8ve
| [[125/63]], ''[[160/81]]'', [[195/98]], [[336/169]]
| Down 8ve
| v8
| v8
| vD
| vD
| Down 10th
| v10
| Jv
|
|-
|-
| 84
| 84
| 1200.000
| 1200.0
| [[2/1]]
| Perfect 8ve
| Perfect 8ve
| P8
| P8
| D
| D
| Perfect 10th
|}
| P10
<nowiki/>* As a 2.3.5.7.13.19.23-subgroup temperament
 
== Notation ==
=== Ups and downs notation ===
 
84edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Ups and downs sharpness|84}}
 
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7|84}}
 
=== 4L 5s (gramitonic) notation ===
This notation is based on Orwell[9]. Notes are denoted as {{nowrap|LsLsLsLss {{=}} JKLMNOPQRJ}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 3 steps in this instance, is denoted by &amp;&nbsp;("amp") and @&nbsp;("at").
 
{| class="wikitable center-1 right-2 center-3"
|-
! #
! Cents
! Note
! Name
! Associated Ratio
|-
| 0
| 0.0
| J
| J
| [[2/1]] exact
| Perfect 0-gramstep
| 1/1
|-
| 8
| 114.3
| K@
| Minor 1-gramstep
| 15/14~16/15
|-
| 11
| 157.1
| K
| Major 1-gramstep
| 11/10~12/11
|-
| 16
| 228.6
| L@
| Diminished 2-gramstep
| 8/7
|-
| 19
| 271.4
| L
| Perfect 2-gramstep
| 7/6
|-
| 27
| 385.7
| M@
| Minor 3-gramstep
| 5/4
|-
| 30
| 428.6
| M
| Major 3-gramstep
| 9/7
|-
| 35
| 500.0
| N@
| Minor 4-gramstep
| 4/3
|-
| 38
| 542.9
| N
| Major 4-gramstep
| 11/8~15/11
|-
| 46
| 657.1
| O@
| Minor 5-gramstep
| 16/11~22/15
|-
| 49
| 700.0
| O
| Major 5-gramstep
| 3/2
|-
| 54
| 771.4
| P@
| Minor 6-gramstep
| 14/9
|-
| 57
| 814.3
| P
| Major 6-gramstep
| 8/5
|-
|-
| 65
| 928.6
| Q@
| Perfect 7-gramstep
| 12/7
|-
| 68
| 971.4
| Q
| Augmented 7-gramstep
| 7/4
|-
| 73
| 1042.9
| R@
| Minor 8-gramstep
| 11/6~20/11
|-
| 76
| 1085.7
| R
| Major 8-gramstep
| 15/8~28/15
|-
| 84
| 1200.0
| J
| Perfect 9-gramstep
| 2/1
|}
|}
== Approximation to JI ==
=== 15-odd-limit intervals ===
{{Q-odd-limit intervals|84}}
=== Higher-limit JI ===
84edo has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], [[31/1|31]], 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit, with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics. In fact, it is [[consistent]] in the no-11 no-17 no-27 no-37 no-47 no-49 no-51 no-55 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~0.13{{cent}} (off by ~7.3{{cent}}), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too when supported by context. The only missing primes are thus 17, 37, 47, 67, 71, 79 and 83, which except for 17 are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 892: Line 769:
| 2.3.5
| 2.3.5
| 78732/78125, 531441/524288
| 78732/78125, 531441/524288
| {{val| 84 133 195 }}
| {{Mapping| 84 133 195 }}
| +0.498
| +0.498
| 0.531
| 0.531
Line 899: Line 776:
| 2.3.5.7
| 2.3.5.7
| 225/224, 1728/1715, 78732/78125
| 225/224, 1728/1715, 78732/78125
| {{val| 84 133 195 236 }}
| {{Mapping| 84 133 195 236 }}
| +0.141
| +0.141
| 0.769
| 0.769
| 5.39
| 5.39
|-
|-
| 2.3.5.7.13
| 225/224, 351/350, 640/637, 1701/1690
| {{Mapping| 84 133 195 236 311 }}
| −0.013
| 0.754
| 5.28
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 441/440, 1344/1331, 1728/1715
| 225/224, 441/440, 1344/1331, 1728/1715
| {{val| 84 133 195 236 291 }} (84)
| {{Mapping| 84 133 195 236 291 }} (84)
| -0.225
| −0.225
| 1.003
| 1.003
| 7.02
| 7.02
|-
|-
| 2.3.5.7.11.13
| 144/143, 225/224, 351/350, 441/440, 975/968
| {{Mapping| 84 133 195 236 291 311 }} (84)
| −0.292
| 0.928
| 6.50
|- style="border-top: double;"
| 2.3.5.7.11
| 2.3.5.7.11
| 99/98, 121/120, 176/175, 78732/78125
| 99/98, 121/120, 176/175, 78732/78125
| {{val|84 133 195 236 290}} (84e)
| {{Mapping| 84 133 195 236 290 }} (84e)
| +0.601
| +0.601
| 1.151
| 1.151
| 8.05
| 8.05
|-
| 2.3.5.7.11.13
| 99/98, 121/120, 176/175, 275/273, 1701/1690
| {{Mapping| 84 133 195 236 290 311 }} (84e)
| +0.396
| 1.146
| 8.02
|}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| rowspan="2" | 1
| 1
| rowspan="2" | 19\84
| 19\84
| rowspan="2" | 271.43
| 271.4
| rowspan="2" | 7/6
| 7/6
| [[Orwell]] (84e)
| [[Orwell]] (84e) / [[newspeak]] (84)
|-
| [[Newspeak]] (84p)
|-
|-
| 1
| 1
| 25\84
| 25\84
| 357.14
| 357.1
| 768/625
| 768/625
| [[Dodifo]]
| [[Dodifo]]
Line 943: Line 841:
| 1
| 1
| 27\84
| 27\84
| 385.71
| 385.7
| 5/4
| 5/4
| [[Mutt]]
| [[Mutt]]
Line 949: Line 847:
| 1
| 1
| 31\84
| 31\84
| 442.86
| 442.9
| 125/81
| 162/125
| [[Sensei]]
| [[Sensei]]
|-
|-
| 1
| 1
| 41\84
| 41\84
| 585.71
| 585.7
| 7/5
| 7/5
| [[Merman]]
| [[Merman]]
Line 961: Line 859:
| 2
| 2
| 5\84
| 5\84
| 71.43
| 71.4
| 25/24
| 25/24
| [[Narayana]]
| [[Narayana]]
Line 967: Line 865:
| 2
| 2
| 11\84
| 11\84
| 157.14
| 157.1
| 35/32
| 35/32
| [[Bison]]
| [[Bison]]
Line 973: Line 871:
| 2
| 2
| 13\84
| 13\84
| 185.71
| 185.7
| 10/9
| 10/9
| [[Secant]]
| [[Secant]]
Line 979: Line 877:
| 3
| 3
| 11\84
| 11\84
| 157.14
| 157.1
| 35/32
| 35/32
| [[Nessafof]]
| [[Nessafof]]
|-
| 6
| 5\84
| 71.4
| 25/24
| [[Trivish]]
|-
|-
| 7
| 7
| 5\84
| 5\84
| 500.00<br>(14.29)
| 14.3
| 4/3<br>(81/80)
| 81/80
| [[Absurdity]]
| [[Absurdity]]
|-
|-
| 12
| 12
| 27\84<br>(1\84)
| 1\84
| 385.71<br>(14.29)
| 14.3
| 5/4<br>(126/125)
| 126/125
| [[Compton]]
| [[Compton]]
|-
|-
|21
| 12
|41\84<br>(1\84)
| 2\84
|585.71<br>(14.29)
| 28.6
|91875/65536<br>(126/125)
| 64/63
|[[Akjayland]]
| [[Catler]] (84c)
|-
| 21
| 1\84)
| 14.3
| 126/125
| [[Akjayland]]
|-
|-
| 28
| 28
| 49\84<br>(1\84)
| 1\84
| 500.00<br>(14.29)
| 14.3
| 4/3<br>(105/104)
| 105/104
| [[Oquatonic]]
| [[Oquatonic]]
|}
|}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Scales ==
== Scales ==
''See also: [[5- to 10-tone scales in 84edo]]''
=== MOS ===
=== MOS ===
Brightest mode is listed.
Brightest mode is listed.


* [[Orwell]]
* [[Orwell]]
** Orwell[9], [[4L 5s]] - 11 8 11 8 11 8 11 8 8  
** Orwell[9] ([[4L 5s]]) – 11 8 11 8 11 8 11 8 8  
** Orwell[13] - [[9L 4s]] - 8 8 8 3 8 8 3 8 8 3 8 8 3
** Orwell[13] ([[9L 4s]]) – 8 8 8 3 8 8 3 8 8 3 8 8 3
** Orwell[22] - [[13L 9s]]
** Orwell[22] ([[13L 9s]])
** Orwell[31] - [[22L 9s]]
** Orwell[31] ([[22L 9s]])
 
=== Other ===
* [[5- to 10-tone scales in 84edo]]
* [[Maeve Gutierrez|Gutierrez Moonglade scale]]


=== Subset of MOS ===
== Instruments ==
* [[Compton]][48]
If you have a precise enough tuner and stable enough instruments, 84edo can be played using 7 instruments tuned a 14th of a tone apart.
** [[Blackened skies]]
** [[Lost spirit]]


=== Other ===
You could also try the [[Lumatone mapping for 84edo]]


== Music ==
== Music ==
Line 1,033: Line 944:
* ''Two5'' for tenor trombone and piano (1991) [https://youtu.be/YOtQZIqfY1w Fulkerson &amp; Denyer recording (YouTube)]
* ''Two5'' for tenor trombone and piano (1991) [https://youtu.be/YOtQZIqfY1w Fulkerson &amp; Denyer recording (YouTube)]
* ''Two6'' for violin and piano (1992) [https://youtu.be/XkX37zH6AbU Haar &amp; Snijders recording (YouTube)]
* ''Two6'' for violin and piano (1992) [https://youtu.be/XkX37zH6AbU Haar &amp; Snijders recording (YouTube)]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/Sqkxrmwggr0 ''microtonal improvisation in 84edo''] (2025)
* [https://www.youtube.com/shorts/Qu6UIA2NmmQ ''84edo groove''] (2026)


; [[Eliora]]
; [[Eliora]]

Latest revision as of 15:15, 20 June 2026

← 83edo 84edo 85edo →
Prime factorization 22 × 3 × 7
Step size 14.2857 ¢ 
Fifth 49\84 (700 ¢) (→ 7\12)
Semitones (A1:m2) 7:7 (100 ¢ : 100 ¢)
Consistency limit 9
Distinct consistency limit 9

84 equal divisions of the octave (abbreviated 84edo or 84ed2), also called 84-tone equal temperament (84tet) or 84 equal temperament (84et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 84 equal parts of about 14.3 ¢ each. Each step represents a frequency ratio of 21/84, or the 84th root of 2.

Theory

84edo shares the perfect fifth with 12edo, tempering out the Pythagorean comma in its patent val. In the 5-limit it tempers out the sensipent comma; in the 7-limit 225/224, 1728/1715, 2430/2401, 6144/6125, supporting orwell, compton, and sensei. In the 13-limit it is the optimal patent val for the rank-5 temperament tempering out 144/143.

84edo is where the orwell temperament takes its name from, since the generator of 7/6 is equal to 19 steps of the edo, referencing the book 1984. Orwell in 84edo comes in two varieties—the 84e val ⟨84 133 195 236 290], supporting the original orwell, and its patent val ⟨84 133 195 236 291] supporting newspeak. 84edo orwell offers mos scales of size 9, 13, 22, and 31, of which the 31-note scale is the maximal evenness scale.

Prime harmonics

Approximation of prime harmonics in 84edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.00 -1.96 -0.60 +2.60 +5.82 +2.33 -4.96 +2.49 +0.30 -1.01 -2.18 +5.80
Relative (%) +0.0 -13.7 -4.2 +18.2 +40.8 +16.3 -34.7 +17.4 +2.1 -7.0 -15.2 +40.6
Steps
(reduced) 		84
(0) 		133
(49) 		195
(27) 		236
(68) 		291
(39) 		311
(59) 		343
(7) 		357
(21) 		380
(44) 		408
(72) 		416
(80) 		438
(18)
Approximation of prime harmonics in 84edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) -0.49 +2.77 +5.92 -2.08 -2.03 -2.60 +6.41 +6.02 +0.78 +6.89 +7.10 +0.55
Relative (%) -3.4 +19.4 +41.5 -14.5 -14.2 -18.2 +44.9 +42.1 +5.5 +48.2 +49.7 +3.8
Steps
(reduced) 		450
(30) 		456
(36) 		467
(47) 		481
(61) 		494
(74) 		498
(78) 		510
(6) 		517
(13) 		520
(16) 		530
(26) 		536
(32) 		544
(40)

Subsets and supersets

84 is a largely composite number. Since 84 factors as 22 × 3 × 7, 84edo has subset edos 2, 3, 4, 6, 7, 12, 14, 21, 28, 42. Being a small multiple of 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.

Intervals

# Cents Approximate ratios* Ups and downs notation
0 0.0 1/1 Perfect 1sn P1 D
1 14.3 81/80, 105/104, 126/125, 169/168, 196/195 Up 1sn ^1 ^D
2 28.6 50/49, 64/63, 65/64, 91/90 Dup 1sn ^^1 ^^D
3 42.9 36/35, 40/39, 46/45, 49/48 Trup 1sn ^^^1 ^^^D
4 57.1 27/26 Trudminor 2nd vvvm2 vvvEb
5 71.4 24/23, 25/24, 26/25, 28/27 Dudminor 2nd vvm2 vvEb
6 85.7 20/19, 21/20 Downminor 2nd vm2 vEb
7 100.0 19/18 Minor 2nd m2 Eb
8 114.3 15/14, 16/15 Upminor 2nd ^m2 ^Eb
9 128.6 14/13 Dupminor 2nd ^^m2 ^^Eb
10 142.9 13/12 Trupminor 2nd ^^^m2 ^^^Eb
11 157.1 23/21 Trudmajor 2nd vvvM2 vvvE
12 171.4 21/19 Dudmajor 2nd vvM2 vvE
13 185.7 10/9 Downmajor 2nd vM2 vE
14 200.0 9/8 Major 2nd M2 E
15 214.3 26/23 Upmajor 2nd ^M2 ^E
16 228.6 8/7 Dupmajor 2nd ^^M2 ^^E
17 242.9 15/13, 23/20 Trupmajor 2nd ^^^M2 ^^^E
18 257.1 52/45 Trudminor 3rd vvvm3 vvvF
19 271.4 7/6 Dudminor 3rd vvm2 vvF
20 285.7 45/38, 46/39 Downminor 3rd vm3 vF
21 300.0 19/16, 25/21, 32/27 Minor 3rd m3 F
22 314.3 6/5 Upminor 3rd ^m3 ^F
23 328.6 23/19 Dupminor 3rd ^^m3 ^^F
24 342.9 28/23, 39/32 Trupminor 3rd ^^^m3 ^^^F
25 357.1 16/13 Trudmajor 3rd vvvM3 vvvF#
26 371.4 26/21 Dudmajor 3rd vvM3 vvF#
27 385.7 5/4 Downmajor 3rd vM3 vF#
28 400.0 24/19 Major 3rd M3 F#
29 414.3 19/15 Upmajor 3rd ^M3 ^F#
30 428.6 9/7, 23/18, 32/25 Dupmajor 3rd ^^M3 ^^F#
31 442.9 84/65 Trupmajor 3rd ^^^M3 ^^^F#
32 457.1 13/10, 30/23 Trud 4th vvv4 vvvG
33 471.4 21/16 Dud 4th vv4 vvG
34 485.7 65/49 Down 4th v4 vG
35 500.0 4/3 Perfect 4th P4 G
36 514.3 27/20 Up 4th ^4 ^G
37 528.6 19/14 Dup 4th ^^4 ^^G
38 542.9 26/19 Trup 4th ^^^4 ^^^G
39 557.1 18/13 Trudaug 4th vvvA4 vvvG#
40 571.4 25/18, 32/23 Dudaug 4th vvA4 vvG#
41 585.7 7/5 Downaug 4th vA4 vG#
42 600.0 27/19, 38/27 Aug 4th, Dim 5th A4, d5 G#, Ab
43 614.3 10/7 Updim 5th ^d5 ^Ab
44 628.6 23/16, 36/25 Dupdim 5th ^^d5 ^^Ab
45 642.9 13/9 Trupdim 5th ^^^d5 ^^^Ab
46 657.1 19/13 Trud 5th vvv5 vvvA
47 671.4 28/19 Dud 5th vv5 vvA
48 685.7 40/27 Down 5th v5 vA
49 700.0 3/2 Perfect 5th P5 A
50 714.3 98/65 Up 5th ^5 ^A
51 728.6 32/21 Dup 5th ^^5 ^^A
52 742.9 20/13, 23/15 Trup 5th ^^^5 ^^^A
53 757.1 65/42 Trudminor 6th vvvm6 vvvBb
54 771.4 14/9, 25/16, 36/23 Dudminor 6th vvm6 vvBb
55 785.7 30/19 Downminor 6th vm6 vBb
56 800.0 19/12 Minor 6th m6 Bb
57 814.3 8/5 Upminor 6th ^m6 ^Bb
58 828.6 21/13 Dupminor 6th ^^m6 ^^Bb
59 842.9 13/8 Trupminor 6th ^^^m6 ^^^Bb
60 857.1 23/14, 64/39 Trudmajor 6th vvvM6 vvvB
61 871.4 38/23 Dudmajor 6th vvM6 vvB
62 885.7 5/3 Downmajor 6th vM6 vB
63 900.0 32/19, 27/16, 42/25 Major 6th M6 B
64 914.3 39/23, 76/45 Upmajor 6th ^M6 ^B
65 928.6 12/7 Dupmajor 6th ^^M6 ^^B
66 942.9 45/26 Trupmajor 6th ^^^M6 ^^^B
67 957.1 26/15, 40/23 Trudminor 7th vvvm7 vvvC
68 971.4 7/4 Dudminor 7th vvm7 vvC
69 985.7 23/13 Downminor 7th vm7 vC
70 1000.0 16/9 Minor 7th m7 C
71 1014.3 9/5 Upminor 7th ^m7 ^C
72 1028.6 38/21 Dupminor 7th ^^m7 ^^C
73 1042.9 42/23 Trupminor 7th ^^^m7 ^^^C
74 1057.1 24/13 Trudmajor 7th vvvM7 vvvC#
75 1071.4 13/7 Dudmajor 7th vvM7 vvC#
76 1085.7 15/8, 28/15 Downmajor 7th vM7 vC#
77 1100.0 36/19 Major 7th M7 C#
78 1114.3 19/10, 40/21 Upmajor 7th ^M7 ^C#
79 1128.6 23/12, 25/13, 27/14, 48/25 Dupmajor 7th ^^M7 ^^C#
80 1142.9 52/27 Trupmajor 7th ^^^M7 ^^^C#
81 1157.1 35/18, 39/20, 96/49 Trud 8ve vvv8 vvvD
82 1171.4 45/23, 49/25, 63/32, 128/65, 180/91 Dud 8ve vv8 vvD
83 1185.7 125/63, 160/81, 195/98, 336/169 Down 8ve v8 vD
84 1200.0 2/1 Perfect 8ve P8 D

* As a 2.3.5.7.13.19.23-subgroup temperament

Notation

Ups and downs notation

84edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.

Semitones 0 1⁄7 2⁄7 3⁄7 4⁄7 5⁄7 6⁄7 1 1 1⁄7 1 2⁄7 1 3⁄7 1 4⁄7 1 5⁄7 1 6⁄7 2 2 1⁄7
Sharp symbol   
 
  
   
     
    
   
  
   
    
     
     
    
   
  
   
Flat symbol
 
  
   
     
    
   
  
   
    
     
     
    
   
  
   

Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:

Semitones 0 1⁄7 2⁄7 3⁄7 4⁄7 5⁄7 6⁄7 1 1+1⁄7 1+2⁄7 1+3⁄7 1+4⁄7 1+5⁄7 1+6⁄7 2 1+1⁄7 1+2⁄7 1+3⁄7
Sharp symbol  
Flat symbol

4L 5s (gramitonic) notation

This notation is based on Orwell[9]. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & ("amp") and @ ("at").

# Cents Note Name Associated Ratio
0 0.0 J Perfect 0-gramstep 1/1
8 114.3 K@ Minor 1-gramstep 15/14~16/15
11 157.1 K Major 1-gramstep 11/10~12/11
16 228.6 L@ Diminished 2-gramstep 8/7
19 271.4 L Perfect 2-gramstep 7/6
27 385.7 M@ Minor 3-gramstep 5/4
30 428.6 M Major 3-gramstep 9/7
35 500.0 N@ Minor 4-gramstep 4/3
38 542.9 N Major 4-gramstep 11/8~15/11
46 657.1 O@ Minor 5-gramstep 16/11~22/15
49 700.0 O Major 5-gramstep 3/2
54 771.4 P@ Minor 6-gramstep 14/9
57 814.3 P Major 6-gramstep 8/5
65 928.6 Q@ Perfect 7-gramstep 12/7
68 971.4 Q Augmented 7-gramstep 7/4
73 1042.9 R@ Minor 8-gramstep 11/6~20/11
76 1085.7 R Major 8-gramstep 15/8~28/15
84 1200.0 J Perfect 9-gramstep 2/1

Approximation to JI

15-odd-limit intervals

The following tables show how 15-odd-limit intervals are represented in 84edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 84edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/7, 14/13 0.273 1.9
5/4, 8/5 0.599 4.2
5/3, 6/5 1.356 9.5
3/2, 4/3 1.955 13.7
13/8, 16/13 2.329 16.3
15/8, 16/15 2.554 17.9
7/4, 8/7 2.603 18.2
13/10, 20/13 2.929 20.5
7/5, 10/7 3.202 22.4
11/7, 14/11 3.222 22.6
9/5, 10/9 3.311 23.2
13/11, 22/13 3.495 24.5
9/8, 16/9 3.910 27.4
13/12, 24/13 4.284 30.0
11/9, 18/11 4.551 31.9
7/6, 12/7 4.558 31.9
15/13, 26/15 4.884 34.2
15/14, 28/15 5.157 36.1
11/8, 16/11 5.825 40.8
15/11, 22/15 5.906 41.3
13/9, 18/13 6.239 43.7
11/10, 20/11 6.424 45.0
11/6, 12/11 6.506 45.5
9/7, 14/9 6.513 45.6
15-odd-limit intervals in 84edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/7, 14/13 0.273 1.9
5/4, 8/5 0.599 4.2
5/3, 6/5 1.356 9.5
3/2, 4/3 1.955 13.7
13/8, 16/13 2.329 16.3
15/8, 16/15 2.554 17.9
7/4, 8/7 2.603 18.2
13/10, 20/13 2.929 20.5
7/5, 10/7 3.202 22.4
11/7, 14/11 3.222 22.6
9/5, 10/9 3.311 23.2
13/11, 22/13 3.495 24.5
9/8, 16/9 3.910 27.4
13/12, 24/13 4.284 30.0
7/6, 12/7 4.558 31.9
15/13, 26/15 4.884 34.2
15/14, 28/15 5.157 36.1
11/8, 16/11 5.825 40.8
13/9, 18/13 6.239 43.7
11/10, 20/11 6.424 45.0
9/7, 14/9 6.513 45.6
11/6, 12/11 7.780 54.5
15/11, 22/15 8.379 58.7
11/9, 18/11 9.735 68.1

Higher-limit JI

84edo has fairly good approximation to higher prime harmonics such as 13, 19, 23, 29, 31, 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit, with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics. In fact, it is consistent in the no-11 no-17 no-27 no-37 no-47 no-49 no-51 no-55 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~0.13 ¢ (off by ~7.3 ¢), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too when supported by context. The only missing primes are thus 17, 37, 47, 67, 71, 79 and 83, which except for 17 are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 78732/78125, 531441/524288 [⟨84 133 195]] +0.498 0.531 3.72
2.3.5.7 225/224, 1728/1715, 78732/78125 [⟨84 133 195 236]] +0.141 0.769 5.39
2.3.5.7.13 225/224, 351/350, 640/637, 1701/1690 [⟨84 133 195 236 311]] −0.013 0.754 5.28
2.3.5.7.11 225/224, 441/440, 1344/1331, 1728/1715 [⟨84 133 195 236 291]] (84) −0.225 1.003 7.02
2.3.5.7.11.13 144/143, 225/224, 351/350, 441/440, 975/968 [⟨84 133 195 236 291 311]] (84) −0.292 0.928 6.50
2.3.5.7.11 99/98, 121/120, 176/175, 78732/78125 [⟨84 133 195 236 290]] (84e) +0.601 1.151 8.05
2.3.5.7.11.13 99/98, 121/120, 176/175, 275/273, 1701/1690 [⟨84 133 195 236 290 311]] (84e) +0.396 1.146 8.02

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 19\84 271.4 7/6 Orwell (84e) / newspeak (84)
1 25\84 357.1 768/625 Dodifo
1 27\84 385.7 5/4 Mutt
1 31\84 442.9 162/125 Sensei
1 41\84 585.7 7/5 Merman
2 5\84 71.4 25/24 Narayana
2 11\84 157.1 35/32 Bison
2 13\84 185.7 10/9 Secant
3 11\84 157.1 35/32 Nessafof
6 5\84 71.4 25/24 Trivish
7 5\84 14.3 81/80 Absurdity
12 1\84 14.3 126/125 Compton
12 2\84 28.6 64/63 Catler (84c)
21 1\84) 14.3 126/125 Akjayland
28 1\84 14.3 105/104 Oquatonic

* In minimal-generator form

Scales

MOS

Brightest mode is listed.

Other

Instruments

If you have a precise enough tuner and stable enough instruments, 84edo can be played using 7 instruments tuned a 14th of a tone apart.

You could also try the Lumatone mapping for 84edo

Music

John Cage
Bryan Deister
Eliora
JUMBLE
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